Majid Szizi and his 12-year-old brother walk wearily, hand in hand, along a railway track from Serbia into Hungary, through the gap in its razor-wire fence. Szizi, 20, from a town in western Syria, said: “Our parents are in Syria. We are going to Denmark where our older brother is living.”

As they make their way, four other young men run into the tall wheat fields nearby, away from the police who have rounded up several hundred migrants 500 metres ahead, and towards the people traffickers in small Suzukis parked up on the road. A Syrian aged about 30 who has just crossed the border prays to the east next to the railway track, kneeling on a sleeping bag.

Hungary has become the new focal point of Europe’s migration crisis, with more than 2,000 people – chiefly refugees from Syria and Afghanistan – now arriving daily. On Tuesday, Hungarian border police apprehended a record 2,330 migrants, mainly Syrians, Afghans and Pakistanis, in the border county of Csongrád, of whom 555 were children.

The exodus appears unstoppable: thousands more streamed through the gap in the railway fence on Wednesday. Aware that the barrier is increasingly at risk of being seen as another failed vanity project from the prime minister, Viktor Orbán, police officers at the scene shooed away reporters. One of them said: “My job today is to make you go away.”

But the fence appears to have had the opposite effect, instead encouraging migrants to accelerate towards Hungary’s Schengen border. In the early hours of Wednesday, hundreds of people took bolt cutters to the fence. Others used blankets or sleeping bags to cover it and jumped over into the fields towards the city of Szeged, about 10 miles away.

Orbán has also been accused of seeking to co-opt the policies of Hungary’s far right Jobbik party. It was a local mayor with close links to the movement who first floated the idea of a border fence in the autumn. Later on Wednesday, Jobbik’s leader, Gábor Vona, said he would visit the border village of Röszke – home to an overcrowded temporary reception centre – to survey the situation.

That situation is increasingly fraught. Hungarian police teargassed about 300 migrants at the 1,100-capacity processing facility on Wednesday. Several hundred Afghan refugees were involved in a brief scuffle with police against the prolonged registration procedure and fingerprinting at the centre, where people crossing over into the EU are held for 36 hours before being transferred to a centre elsewhere in Hungary.

Many of these migrants had spent several days in a bottleneck on the Greek-Macedonian border last week, when the latter country declared a state of emergency for several days before lifting the declaration on Sunday.

A local police official, Szabolcs Szenti, dismissed the chaotic scenes as a little incident and said the problem now unfolding in southern Hungary was a national one. Behind him, a van containing Romanian officials from Frontex, the EU’s border coordination agency, and a Dutch police van entered the facility. Several other coaches left, each carrying about 60 migrants, mainly families.

Balázs Szalai, of the migrant solidarity group Migszol Szeged said the police had been overwhelmed and uncoordinated, leaving much of the humanitarian work to the civil sector. He said: “We cleaned up the litter from the main migrant waiting area this morning, when a local police chief also told us we can go with the migrants to help. We were helping, but now we are being told to go away.”

Szalai said the fence is unnecessary: “They should not be building a fence and organising billboard campaigns. They could be using resources to manage this situation, like building smaller camps so that children are not at risk of being teargassed.”

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Leila Mirzaei, a final year medical student from Tehran, has been helping with treatments and translation issues at Migszol’s migrant help centre in Szeged. She said: “The most common are abscesses from animal or insect bites. Gastric problems, colds and flu are also common. With children, fever is also common.” Mirzaei pays for the drugs herself.

Despite the manifold obstacles, the migrants keep coming. Late on Wednesday, Jusef Maen, 37, walked north along the railway tracks into Hungary with his wife and children, aged three and four. He said: “We travelled from Homs in Syria to Damascus, then Kalymnos, Athens, Macedonia and Serbia, using plane, train, walking, bus and taxi.”

The migrants stay up to date with the latest legal information via smartphones and are reluctant to be fingerprinted in Hungary. Maen, who describes himself as a T-shirt and trouser merchandiser, said: “If I get fingerprinted we will go to Stockholm, if not Berlin. I spent $9,000 on this trip, but now I have nothing.” He shrugs. “Allah”, he says, pointing towards the sky.